1. From: Space-by-time manifold representation of dynamic facial expressions for emotion categorization
Journal of Vision. 2016;16(8):14. doi: /
Figure Legend:
Illustrative application of our method to simulated AU data. (A) Comparison of dimensionality-reduction methods. Here we show how three dimensionality reduction methods (NMF, color-coded in orange; PCA, color-coded in cyan; and ICA, color-coded in magenta) recode a simulated data set comprising the activations of two AUs—Upper Lid Raiser (AU5) on the y-axis and Jaw Drop (AU26) on the x-axis. Data are generated by the linear combination of two basis functions w1 and w2 that represent two functional dimensions. The first dimension consists of a low activation of Upper Lid Raiser and a high activation of Jaw Drop; the second, vice versa (see bars). Gray spheres reflect individual trials (600 in total) represented with different coefficients (c1 and c2) for the basis functions. On half of the trials, we impose correlations (p = 0.7) between c1 and c2 to represent AU synergies. NMF recovers the correct basis functions (see orange axes). In contrast, neither PCA (cyan axes) nor ICA (magenta axes) recovers the original basis, because of their underlying assumptions. PCA starts from the dimension explaining the most variance, adding one orthogonal dimension, and ICA looks for independent dimensions. For both PCA and ICA, the second dimension comprises negative values for one of the two AUs, which is incompatible with their functional role as representations of AU activations. (B) Emotion discrimination in the NMF space. In the space defined by w1 and w2, we apply LDA to discriminate trials categorized as fear (blue spheres) from those categorized as surprise (red spheres). LDA determines the categorization boundary (green line) that reliably discriminates the two emotions (93% correct discrimination).
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